Does hair always start growing 3 months after a hair transplant ?

Hair regrowth is variable

Hair growth starts about 3 months after the surgery. But keep in mind it's highly variable - for some it's 2 months and some it's 5. 

But regrowth time also depends on the type of surgery. We see some delay in patients with scarring alopecias (cicatricial alopecia), compared to those with genetic hair loss. 

But overall, three months is a safe bet.